yunfan cai detailed numerical stimulation of experiments on masonry arch … · 2020. 2. 12. · on...

Yunfan Cai

D e t a i l e d n u m e r i c a lstimulation of experimentson masonry arch bridgesby using 3D FE

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Yunfan CaiDetailed numericalstimulation of experimentson masonry arch bridgesusing 3D FE

Barcelona 2011

Detailed numerical stimulation of experiments on masonry arch bridges using 3D FE

Erasmus Mundus Programme

ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS i

DECLARATION

Name: Yunfan Cai

Email: [email protected]

Title of the

Msc Dissertation:

Detailed numerical stimulation of experiments on masonry arch bridge using 3D

FE

Supervisor(s): Climent Molins

Year: 2011

I hereby declare that all information in this document has been obtained and presented in accordance

with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I

have fully cited and referenced all material and results that are not original to this work.

I hereby declare that the MSc Consortium responsible for the Advanced Masters in Structural Analysis

of Monuments and Historical Constructions is allowed to store and make available electronically the

present MSc Dissertation.

University: UPC

Date: July 27th

Signature:



ii ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS



ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS iii

ACKNOWLEDGEMENTS

This dissertation would not be completed without the help of all that, directly or indirectly, were involved in my work and successfully contributed to its conclusion. Among the different support, I would like to express my gratitude to:

• Professor Climent Molins, my supervisor and mentor, for his full support and encouragement, the interesting discussions, patience and careful instruction;

• PhD student Oriol Arnau and Ahmed Elyamani, for their tremendous help and support in

the modeling in DIANA and many ideas of the dissertation;

• the Erasmus Mundus Committee and European Union, for fully support me with the scholarship to finish the master study; Finally, I would like to make a special acknowledgment to my family:

• Ying Sun, my mother, for her kindness and encourage during this four month, and most importantly, the support for me in every difficult times;

• Haqi Cai, my father, for all he did to make my dream of studying abroad come true, and offer me a great environment to study and research;

•All the other family members that paid so much care and attention on me. You are the source of the power for me to move on.



iv ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS



ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS v

ABSTRACT

The constructions of masonry arch bridge had been found 4000 years ago and until now they are still adopted in many areas. So it is of great importance to understand the structural behavior of different components in order to preserve them in good service conditions.

The bibliographic research is made first to be aware of the state of art on masonry arch bridge, which include basic behavior of masonry arch bridge, development of analysis method, collapse mechanism and experiment campaign of previous times. In this dissertation, the experimental study which was carried out by Pere Roca (2001) about masonry arch bridge was briefly summarized. Then, the limit analysis of collapse mechanisms and ultimate capacity was used for a first behavior assessment.

After that, plane stress numerical models were prepared using finite elements and it was based on the experimental results. Comparison of the results obtained from different finite element models were made, they are: (1) one soil infill layer with steel bracing system; (2) one soil infill layer without bracing system; (3) three soil infill layers with steel bracing system; (4) three soil infill layers and steel bracing system with masonry tensile strength 0.02, 0.05 and 0.08 N/mm2 respectively. As final step, three dimensional finite element analysis with interface element was carried out aiming to consider the effect of spandrel walls and the realistic experimental environment.



vi ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS



ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS vii

RESUMEN

Las construcciones de puente de arco de obra fábrica han sido utilizadas desde hace 4000 años hasta la actualidad siguen siendo usadas como alternativa en muchas áreas. Por lo tanto, es de suma importancia entender el comportamiento estructural de los distintos componentes de estos con el fin de mantenerlos en buenas condiciones.

Una bibliográfica se hizo en primer lugar para conocer el estado del arte en puente de arco de obra de fábrica, que incluyen el comportamiento básico del material del puente de arco, el desarrollo del método de análisis, el mecanismo de colapso y de los experimentos previamente realizados. En esta tesina, se resumió brevemente el estudio experimental que se llevó a cabo por Pere Roca (2001) sobre puente de arco de obra de fábrica. A continuación, el análisis límite de mecanismos de colapso y de la capacidad máxima se utilizó inicialmente para la evaluación del comportamiento.

Después de eso, los modelos de tensión plana fueron preparados utilizando elementos finitos basándose en los resultados experimentales. Se realizaron ccomparaciones de resultados obtenidos a partir de diferentes modelos de elementos finitos se realizaron, son: (1) una capa de relleno del suelo con refuerzo de acero, (2) una capa de relleno de suelo sin refuerzo, (3) tres capas de relleno del suelo con el refuerzo de acero , (4) tres capas de suelo de relleno y un refuerzo de acero con una resistencia a la tracción del bloque de albañilería de 0,02, 0,05 y 0,08 N/mm2, respectivamente. Como paso final, se llevó a cabo el análisis tridimensional de elementos finitos con elementos de la interfaz con el objetivo de considerar el efecto de las paredes del tímpano y el entorno experimental real.



viii ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS



ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS ix

摘要

砌体结构拱桥的建造可以追溯到4000年前，并且直到现在还广泛应用与各个领域。因此，

了解各个结构部件的性能以至于能够完好地保护他们就变得十分重要。

首先，本文首先收集了关于砌体结构拱桥的基本性能、计算方法的发展演变、破坏机制

和前人试验情况的文献资料并进行了阐述。在本论文中，简要介绍了Pere Roca教授在

2001年进行的砌体结构拱桥的试验研究。然后应用极限分析法作为第一步的性能分析

评价破坏机制和极限荷载。

其后，基于试验结果，建立了平面应力有限元模型。并且对于不同模型的结果进行了比

较和分析，他们包括：（1）带钢支撑的，一层填充土层的模型；（2）不带钢支撑的，

一层填充土层的模型；（3）带钢支撑的，三层填充土层的模型；（4）带钢支撑的，三

层填充土层，砌体抗拉强度分别为 0.02, 0.05 和0.08 N/mm2 的模型。最后，针对拱

肩墙和真是试验环境的模拟，建立了三维有限元模型并进行分析。



x ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS



ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 1

TABLE OF CONTENTS 1. INTRODUCTION .......................................................................................................................1

1.1 A brief overview of masonry arch bridge ......................................................................................... 1

1.2 Example of masonry arch bridge structures in China .................................................................... 2

1.3 Scope and objective............................................................................................................................. 4

1.4 Outline of the thesis ............................................................................................................................ 5

2. CURRENT STATE OF KNOWLEDGE...................................................................................6

2.1 Behavior of a masonry arch bridge ................................................................................................... 6

2.2 Introduction to the methods of analysis of masonry arch bridges ................................................. 9

2.3 Identification of failure modes of masonry arch bridges .............................................................. 14

2.3.1. Experiment ................................................................................................................................ 14

2.3.2. Mechanism of collapse and theories ........................................................................................ 17

3. EXPERIMENT TEST, RESULT AND ANALYSIS ..............................................................20

3.1 Construction of the bridge arch in the laboratory......................................................................... 20

3.1.1. Geometric description of the bridge........................................................................................ 20

3.1.2 Materials used in construction.................................................................................................. 21

3.1.3 Bridge construction process ...................................................................................................... 22

3.2 Implementation of the bridge........................................................................................................... 25

3.3 Conduct of the experiment ............................................................................................................... 27

3.3.1 Process of implementation of the load...................................................................................... 27

3.4 Analysis of test results ...................................................................................................................... 29

3.4.1 Data on the load applied............................................................................................................ 29

3.4.2 Results obtained from gauges embedded in the infill. ............................................................ 30

3.4.4 Results obtained with the load cells.......................................................................................... 36

3.5 Analytical prediction using RING 1.5 ............................................................................................. 37

3.5.1 General information .................................................................................................................. 37

3.5.2 Results and analysis ................................................................................................................... 38

4. FINITE ELEMENT ANALYSIS OF PLANE STRESS MODEL FOR MASONRY BRIDGE .........................................................................................................................................40

Detailed numerical stimulation of experiments on masonry arch bridge using 3D FE


2 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS

4.1 Introduction .......................................................................................................................................40

4.2 Geometry definition...........................................................................................................................41

4.3 Element type.......................................................................................................................................42

4.3.1 Plane stress element....................................................................................................................42

4.3.2 Interface element ........................................................................................................................44

4.3.3 Truss element ..............................................................................................................................44

4.4 Material properties............................................................................................................................45

4.5 Meshing, Boundary conditions and Load........................................................................................48

4.6 Results and analysis...........................................................................................................................51

4.6.1 Linear analysis results................................................................................................................51

4.6.2 Nonlinear analysis results ..........................................................................................................52

4.6.3 Comparison with finite element model of different masonry tensile strength ......................58

4.6.4 Comparison with finite element model of different soil layers...............................................59

4.6.5 Comparison with finite element model without steel bracing system....................................61

5. FINITE ELEMENT ANALYSIS OF THREE DIMENSIONAL MODEL FOR MASONRY ARCH BRIDGE .............................................................................................................................64

5.1 Introduction .......................................................................................................................................64

5.2 Basic assumptions..............................................................................................................................64

5.3 Modeling process ...............................................................................................................................64

5.4 Linear analysis ...................................................................................................................................68

5.5 Nonlinear analysis .............................................................................................................................72

6. SUMMARY AND CONCLUSIONS ........................................................................................77

6.1 Conclusions ........................................................................................................................................77

6.2 Suggestions for further study ...........................................................................................................78

7. REFERENCES...........................................................................................................................80




1. INTRODUCTION

1.1 A brief overview of masonry arch bridge

Due to the aggressions of the environment (snow, rain, cold, heat, etc.) human being started to protect themselves by using the natural materials such as natural caves, tree trunks, animal’s fur, straw, clay, etc. With the appearance of the first civilization, around 9000 to 7000 BC, the construction techniques evolved stone, adobe, wood and clay brick begun to be used.

Archaeological remains of stone arch bridges date back to the Sumerian civilization in

Mesopotamia, around 2000 B.C. The Sumerians assembled stones in the shape of an arch

allowing them to work in compression rather than in bending as a beam bridge. These arch bridges

were not the typical radial arrangement of stone segments, but more of a false arch (corbel arch)

composed of cantilevered brick or stone progressively jutting out. Possibly the oldest existing arch

bridge is the Mycenaean Arkadiko bridge (Figure 1-1) in Greece built around 1300 B.C. Although

the arch was already discovered and known by the Etruscans and Greeks, the Romans were the

first to fully utilize the potential of arches in bridge construction. The Romans turned the masonry

arch into the almost universal method of bridge construction until the 18th century. Many of these

bridges are still standing today and even used for modern traffic loads.

Figure 1-1 Arkadiko Bridge (photo by: David Gavin)

Stone and brick masonry is, effectively, one of the finest and most durable construction techniques ever invented by human. Masonry consists of building stable bonded stacks of small pieces by hand (Vekey, 1998). Used since the time of the first villages and cities built by human, masonry application has been growing and evolving to new uses all over the entire civilized world. It was a fundamental building material in Mesopotamian, Egyptian and Roman periods. During Roman




period, the use of masonry increased and become specialized in order to maximize its benefits. Despite several modifications of the masonry uses, shape and manufacture along thousands of years of constant evolution, the simplicity that made its success remained.

The traditional Roman arches remained the main form of bridge construction for many centuries. However, advances in technology, and better understanding of the forces acting in an arch led to the use of different arch shapes in bridge construction. Many arch shapes that were developed were not used in bridge construction, however, because their shape did not improve the functionality of a bridge. For example, the lancet, or Gothic arch (Figure 1-2 a), was developed in the 12th century and used in tall structures such as cathedrals where their narrow, high pointed shape was efficient in transferring forces to the foundations without large buttressing. This is not needed in a bridge, where the desirable shape is one that allows a long span, rather than a high rise.

Figure 1-2 Common arch shapes in masonry bridges; (a) Lancet, or Gothic; (b) Elliptical; (c) Three-centered; (d) Catenary; (e) Tudor

1.2 Example of masonry arch bridge structures in China

China has a vast territory with many networks of rivers. Throughout the history, the Chinese nation has erected thousands of bridges. Among them, some stone arch bridges survived to today, such as the ChawZhou Bridge.

The steel and cement productivity were short after the establishment of the nation. According to the principle of self-reliance and procuring materials available locally, a lot of stone arch bridges had




been built in highway construction in 1950-1970s. Even at present, they are also used in hilly or mountainous areas. The Huanghugang Bridge with a span of 60m was opened in Hunan Province in 1959. It firstly broke the longest span record in the Chinese arch bridge history after the Chawzhou Bridge. After that a march on large span stone arch bridges was prompted forward. There are more than ten bridges with span length beyond 100m. Such as: Long Rainbow Bridge (the span of 112.5m) in Yunnan Province, 1961; Jiuxigou Bridge (the span of 116m) in Chongqing City, 1972, etc. Since 1970 the newborn two-way curved arch bridges get popular. The development of stone arch bridge stepped down. While in 1991, an all-open spandrel stone rib arch bridge with span of 120m was built in Fenghuang County, Hunan Province. The Danhe River Bridge which is an all-open spandrel ribbed barrel stone bridge was also opened in Shanxi Province in 2000. Its main span is as large as 146m and deck width of 24.8 m (Figure 1-3). The thickness of section is 250 cm at the arch crown and 350 cm at the arch spring. It ranks as the world’s longest span stone arch bridge. Figure 1-4 shows some masonry bridges with long span in China. However, stone arch is completely unstable until the arch rib is enclosed. Therefore, it should be built by elaborating scaffolding, or "centering," below the spans to support them during arch rib construction. In many cases, it is not an economic bridge type to be considered. After 1990s, only four stone arch bridges with a span larger than 100 m have been built.

Figure 1-3 New Danhe Bridge in Shanxi




Figure 1-4 Long span stone arch bridge: (a) Chawzhou Bridge, Hebei Provinces, span of 37.5m, 606, (b) Huanghugang Bridge, Hunan Province, span of 60m, 1960, (c) Jiuxigou Bridge, Chongqing

City, span of 116m, 1972, (d) Danhe Bridge, Shangxi Province, span of 146m, 2000

On the other hand, many stone arch bridges are still in service in the road system. Long service time, poor conditions and increasing traffic are making more and more harm to the existing masonry arch bridges. The maintenance, rehabilitation and strengthening of these stone arch bridges have became a very important topic in China. Bridge management authority and engineers should pay enough attention to these issues and learn from research results and engineering experience practiced in other countries. Just in the proceedings of past international conferences on arch bridges, there are many papers focusing on these aspects.

1.3 Scope and objective

It is shown previously that masonry arch bridge has a long history and is still been used nowadays. In the past centuries, many researches have been carried out to investigate the behavior of masonry arch bridges. Different theories had been developed for design and calculate the masonry arch. Despite disadvantages, many hypothesis are quite useful and being widely applied in different territories. One of the method recently be made use of most is the finite element method. The main




purpose of this research work is to stimulate the experiment of masonry arch bridge using numerical method as limit analysis and finite element analysis.

The main objective of the dissertation is as follows:

Study the structural behavior of masonry arch bridge and analysis method developed in the past.

Understand the experiment of masonry arch bridge (test in Laboratory of structural technology of UPC) in geometry, material used and their properties, construction and loading procedure and results from the experiment.

Limit analysis as a first step to stimulate the bridge using RING 1.5

Set up a suitable finite element model based on the hierarchical nature of the soil and nonlinear behavior of masonry and infill.

Compare the finite element results with different assumption of masonry material properties and boundary conditions based on the verified finite element model; compare the finite element results with the available analytical solution.

Evaluate steel bracing system’s contribution to ultimate capacity and collapse mechanism.

1.4 Outline of the thesis

The outline of thesis is as following. A brief introduction of masonry arch bridge is shown in Chapter 1. Current state of knowledge on masonry arch bridge behavior and calculation as well as the loading characteristic and collapse mechanism is shown in Chapter 2. In Chapter 3, the description on segmental masonry arch bridge experiment is presented. And after the experimental results, a limit analysis using RING is made to stimulate the experimental process in this chapter. Followed to the limit analysis, the 2D finite element analysis is developed, and parametric study based on various masonry properties is shown in chapter 4. three dimensional finite element analysis is taken for a further research on the masonry arch bridge behavior in chapter 5. Summary, conclusion and suggestion for further study is shown in Chapter 6.




2. CURRENT STATE OF KNOWLEDGE

2.1 Behavior of a masonry arch bridge

A masonry arch bridge has a complex structural behavior. Masonry elements interact with each other, together with the infill and the loads applied. There are three fundamental aspects in the resistant behavior of these structures (Martin-Caro, 2001):

- They are massive structures that work primarily by dimensions. The main resistance element is predominant by, in principle, the axial force.

- They are composed of heterogeneous materials, anisotropic and even discontinuous, i.e. masonry, which can not resist tensile stresses.

- Structural elements that are of a different nature and their structural action is also different (arch, infill, spandrel wall, etc.). These structural elements have interaction with each other.

Figures 2-1 and 2-2 are a masonry arch bridge which subjected to the action of self-weight and dead load concomitantly with a uniform load applied along its length and a load consisting of three loading point P, separated by a length L1 between them.

These figures try to show what is the mechanism developed by a bridge and how to disperse the loads from loading point to the foundation. The following are the most important aspects of this mechanism:

It can be seen in these figures that the dead load (weight + dead load) applies directly on the arch and load is transmitted from the road surface to the arch through the infill. Correctly assessing the force transmission path and the area where it is applied is very important. A wider distribution of the load in the cone area affects the behavior of an arch.

Once the loads reach the arch, it is responsible to collect them and bring them to the foundation. In this regard, the definition of the real start of the arch and its junction with the foundation is also deciding the behavior of these structures. Sometimes when talking about infill, the area near the starts of the arch is performed like cemented infill. And this makes the real springing of the arch rise to approximately the level where this infill behavior softer.

Another aspect also of great importance is the definition of the boundary conditions of the arch. Figure 2-1 is considered that the vault is fixed-ended and therefore the set of cemented infill and foundation must take a horizontal force, vertical force and bending moment. But this embedding of the arch is not always present.




Therefore, the support conditions of arch (geometric definition and condition of embedding) must be studied and detailed analysis.

Figure 2-1 Schematic of longitudinal resistance in a masonry arch bridge (Gutierrez, 2001)

Loose infill is located between the arch, spandrel walls and abutment, not just lies over the arch, but also exerts a non-negligible horizontal force on the arch and spandrel walls. Then, these elements act as supporting structures, the longitudinally and transversely spandrel walls.

The thrust of the infill on the arch depends on the relative movements between infill and arch. As these movements are usually small (as for massive structures) the thrust developed by the infill is much like the push to the rest. In situations close to collapse, which are the possible mechanisms in these structures, you can observe big movements and deformations (see Figure 2-1).

The spandrel walls must bear the transverse lateral thrust (Figure 2-2) that the infill has applied on them. Moreover, this thrust can cause failure of the spandrel wall and the consequent loss of stability of the bridge. Spandrel walls can provide additional rigidity contributing to bearing capacity a lot. But it is precisely to see the difference in stiffness between arch and spandrel wall, along with the lateral force on these two elements, could cause separate sometimes on them, then they function as independent elements. So this rigidity contribution should not be taken into account unless it can be ensure that a reliable connection between the arch and spandrel wall has done.




Figure 2-2 Schematic of masonry arch bridge cross-section under load (Gutierrez, 2001)

The ring is in charge of collecting the load which transmits from the infill and disperse it to the foundation. There are two perspectives in the analysis of the force dispersion (Gutierrez, 2001):

One way of understanding this load transfer is simply based on the equations of static. It tries to find out the path of the crossing points of the resultant in each section of the arch by equilibrium relationship with external loads, or whatever it is, trying to find out the line of thrust. Figure 2-1 shows the thrust line, this line marks the route of the resultant until the where the arch starts and the foundation. To calculate the thrust line, it is only made use of the equations of balance, then, in principle, there are infinite solutions for indeterminate structure. To answer the question of which the infinite possible solutions is true, we use the theories of upper and lower limits of plasticity, or by adopting theories that allow you to choose a line of thrust after admitting some hypotheses. Using this classical analysis can not obtain any information regarding the stress level, or the deformations and movements in the arch, because the material constitutive equations and compatibility equations are not used.

The other way to understand how this load transfers is to study the arch behavior under the classical theory of structures, i.e. using the equations of equilibrium, constitutive equations and equations of material compatibility. Using this method provides a single solution, and also the information including the forces, deformations and movements of the arch.




In Figure 2-1 the difference between the thrust line and the mass center indicates the eccentricity exists in each of the sections. The gray areas represent the cracked part of the arch where they can not resist tensile stresses, and black colored areas show where the masonry has reached the compressive strength.

2.2 Introduction to the methods of analysis of masonry arch bridges

The first major studies on arch bridges of masonry were made by Hooke in the XVII century, which were based on the polygon antifunicular. This great English engineer, was the first to realize that the best way to generate the arch should take a smart way to deal with compressive stresses in the arch, writing in 1675: "Hangs the arch it will form a flexible line, invert it will get an arch stand rigidly" (Such as a flexible chain hanging, but reversed, the arch remains stable rigid) (Serna, 2011). Engineers later as Gregory, Poleni, Ware and Fuller moved on research in this direction, however, always end in excessive complexity methods for knowledge and tools restrain of that time.

Therefore, since the early eighteenth century, putting efforts to represent the behavior of these structures, engineers Couplet, Frezier and Coulomb focused more on the limit analysis and conversion of the arch in a collapse mechanism. You could say that this was the beginning of a scientific-technical treatment in the analysis of these structures, combining theoretical studies with experimental tests and observations on failure of these structures.

Later in the nineteenth century, engineers Clapeyron, Navier, Barlow proposed complete analysis methods based on the theory of elasticity. Navier established the law of stress distribution on surface.

Barlow, meanwhile, developed a graphical method for obtaining the thrust line, and based on the work of him, Rankine and Winkler developed various methods of design that, minimizing the length of the thrust line in the profile of the arch. Later, Castigliano presented his work on the minimum strain energy, the first time allowing the calculation of displacements in the structure. But all the elastic methods are inaccurate, especially at failure, since they assumed that the masonry could provide tensile strength, or that the line of thrust was contained in the central third.

Currently, analysis methods can be divided into plastic and elastic methods. The plastic methods based on the formation of plastic hinges on the points needed to reduce the degree of indeterminate structure to transform it into a mechanism. Heynman Pippard was the first and most prominent authors who have followed this path of research.




The elastic analysis of arch bridges of masonry, in turn, has led to a wide range of assessment methods. Pippard was the first to study contrast to experimental observations. His work contributed to determine limit values of W (ultimate load) for parabolic arches given to the ring. Cracking-elastic analysis is an analysis based on classical elastic theory in which the loads are applied in the arch and the stresses are determined. These are used to determine the state of stress, allowing you to identify cracking areas and areas where the material become plastic, which will allow then calculate the deformations and modified geometry. This allows modeling of failure mechanism, and stress variations of the infill.

Later, after the Second World War, the Military Engineering Experimental Establishment in the UK developed a method known as MExE to assess the bearing capacity of masonry bridges. This method is based on the work of Pippard, Pippard modeled the arch as linearly elastic with two pins at the abutments and applied a load at its mid-span (Ford et al 2003). He reasoned that load dispersion through the fill would shift the critical load to the location of the smallest fill depth, the mid-span (Page 1993). MExE introduces modifications of Pippaed’s theory taking into account the longitudinal distribution of the load, the condition of the bridge, ring, infill, height of abutment and other geometric measures.

Already in the eighties and nineties, engineers have developed several specific methods. Bridle and Hughes, for example, raised in 1991 the Cardiff energy methods, based on Castigliano's energy theory, which allows taking into account the material and geometric nonlinearity and the structural impact to the infill.

In recent years, the Finite Element Method has been the type of analysis developed and extended in all types of the structural calculation. Based on the Principle of Virtual Work (PTV), displacements at the nodes is consistent with the boundary conditions imposed, by interpolation. With the PTV it gets balance equation and then tensions. Besides the micro-models (that model of the mortar and blocks separately), which are only useful in laboratory work, the most common is the use of macro model, of which properties are taken with the masonry. At first they were one-dimensional models with elements that took into account the infill with horizontal elements (Crisfield, 1985) in Figure 2-3 shows these horizontal elements that model the infill and the sliding area between the arch and infill (Page , 1993).. Later models with two-dimensional elements were used; more suitable to the study of the infill and are widely used today. Three-dimensional models have a large computational cost and are only justified for very unique case study. Their main problems are the difficulty of reproducing the deformation phenomena that focus on certain points, as is the case of spherical collapse in masonry arch.




Figure 2-3 Horizontal elements that model the infill (Crisfield, 1985)

Then a method that is current focus on research for the good results is based on the generalized nonlinear matrix formulation, combined with simple constitutive models suitable to describe the mechanical behavior of the masonry. The generalized nonlinear matrix formulation, model the structures nodes and bars with quite general geometry and properties, developed guidelines for elements to address the practical limitation posed by different approaches for the final analysis of masonry arch bridges. This limitation is the considerable computational capacity required by the finite element models, in both 2D and 3D combined with appropriate constitutive equations to describe the behavior of the masonry.

It is a method based on continuum mechanics capable of representing the mode very similar to those that predict the ultimate mechanism methods when adopting weak constitutive equations (without traction) to model the masonry under tension. It is also a flexible method based on equilibrium at all points of the given element without additional assumptions on the fields of displacements or strains. In order to carry out the nonlinear analysis of the material, the masonry is considered a material under tension elastoplastic and adopted plastic constitutive equations combined with compression and shear.

The application of the matrix formulation to general nonlinear analysis of masonry arch bridges make it possible to model effectively leverages its elements with target curve. Because the movements are completely free (not in the FEM, where they have to take field trips) and can play important curvatures associated with the damage then you can get an approximation for the location of this damage. Therefore, in this case it does not show numerical problems that appear in the finite element models. The constitutive equations used worked perfectly in combination with the weak formulation of displacement.

The analysis shows the method's ability to predict not only the ultimate load but also the response through the loading process and the mechanism of collapse or, if any, failures by crushing the




masonry before the generation of collapse mechanism. In addition to the method, it allows to include the effect of the infill and the geometric effects of second order.

For the modeling of arch bridges, it is needed to adopt a technique that, using the elements of generalized nonlinear matrix formulation, to simulate not only the arch but also the effect on the stiffness and strength of abutment and spandrel walls. This modeling is shown in the figure below (Roca, 2000):

Figure 2-4 Modelling arch bridges in FMG (Roca, 2000)

Navier pointed out the fact that engineers do not have to be really interested in the ultimate load of the structure (for which Galileo was interested), since this is a state that should be avoided, the main concern is to ensure that the serving state of the structure is safe, and that is achieved by limiting the stress state in a proportion of the limit state. As for that, the limit analysis method had been developed.

Engineers, first of all, must obtain the stresses (internal forces) of structures and the first tools they use are the equations of static, which are only sufficient to isostatic structures, as for statically indeterminate structures there are infinite possibilities of balance. So to find the real state of the structure requires the use of other equations of structural analysis, such as the material constitutive equations and boundary equations.

The two theories that underline the various authors to address the structural analysis of arch bridges are masonry elastic theory and the plastic theory.

In the dissertation of Anàlisi, he developed a computer program based on the method of limit analysis with the plastic theory (Soms, 2001); a program that has been used in this thesis was an approximation of the ultimate load of the bridge had been tested. Based on the four basic assumptions of the plastic-masonry theory has infinite compressive strength (the tension strength of




the masonry is much lower than its compressive strength), the masonry does not support cracking, the masonry is considered a solid deformable (elastic modulus infinity) and no-slip element between consecutive segments. Anàlisi developed a computer program that allows considering the structure of a masonry arch in the ultimate state, with the formation of a collapse mechanism, i.e. assuming the existence of four hinges, which is the number of hinges needed to consider the structure as a mechanism.

The calculation of a structure formed by segments is subjected to rigid components, based on the theory of small deformations and without regarding the movements that take place before the collapse. Therefore, it is likely that these cracks develop along sections of weak part of masonry arches and thus is fully developed considering the spherical collapse mechanisms that transform arches into hinged bar structures, the collapse mechanism shown in the figure below:

Figure 2-5 Collapse mechanism by formation of hinges (Gutierrez, 2001)

Once introduced the four hinges, the ultimate load of collapse P can be determined by the balance and is directly related to the weight of five blocks: W1, W2, W3, W4 and W5, as shown in the figure below (Gutierrez, 2011):




Figure 2-6 Ultimate load from the weight of the 5 blocks

Pippard, in this approach, only considered the contribution of three blocks because it involved the formation of two of the four hinges in the arch starts.

2.3 Identification of failure modes of masonry arch bridges

The type of failure mode and stress on the bridges of masonry arch indicates the ultimate capacity and ductility of the structure. In this section we collect the different failure modes identified from experiments at failure (in both full-scale and in the small-scaled model) and in the history of collapses of these structures. The probability of different modes of failure is different depending on the type of bridge (recessed arches, point arches, etc.).

2.3.1. Experiment

In the second half of the twentieth century a test campaign had been carried out on masonry arch bridges in the UK. Tests have been loaded at failure (scale and reduced model) (Hughes, 1997).




2.3.1.1. Tests at failure

Scale models

There have been 13 experiments at failure on real scale structures. The first three were carried out by Davey before the Second World War, and the rest were organized and financed by the Transport Research Laboratory (TRL). This second part of the campaign began in 1985 and lasted 5 years on 8 existing bridges but disused and of no aesthetic or historical value, and the experiments of two models were in the laboratory. The purpose of this second part of the campaign was reviewing the existing method for evaluating the bearing capacity of these structures, this method was Mexe, and it got the ultimate load for bridge collapse of masonry arch.

It was claimed that the test program covered the largest number of types. It tried to cover different types of materials commonly used in the arch, including concrete blocks, brick, and stone. With only 13 experiments it is difficult to perform a parametric study of many variables accurately and reliably.

The tested was under static loads. Load was applied approximately at one quarter or one third of ring (critical position for this type of bridge). At the loading point, loading beam transversely across the entire width of the arch.

Although the data were obtained during the experiments were not so much, just to figure out the deformation related with the load, but many conclusions were reached.

First we identified potential failure modes in these structures. Moreover, it had been proved the relative importance of different structural elements depending on the bridge type. For example, we tested the infill behavior close to the bridge abutments and influence of the spandrel walls in the failure mode. Thanks to the load-deflection curves and the evolution of the deformation observed, different behaviors of these bridges in service and near failure had been discovered.

The completion of these experiments was a milestone in the understanding of these structures, indicating the way forward in analyzing and identifying the structural elements action and most of the possible collapse modes.

Reduced models

The test for reduced model has drawbacks with respect to the full scale models. In principle, when the failure mode of the structure is not a mechanism, the result is sensitive to certain geometric and




mechanical variables, the scale factor and the model execution. Other difficulties are the action of the infill and the actual state of the structure. In any case, they have conducted nearly 80 essays on small-scale model in which different aspects have been played in two-and three-dimensional behavior of these structures.

The construction of the model to test enables the accurate study of some aspect of a particular behavior or, conversely, the combination of several; well designed experiments have been conducted to study the action of infill, the action of the spandrel walls, and the behavior of multi-arch bridges.

2.3.1.2. Tests in service range

The masonry arch bridges behavior within service range is far from failure. Knowing the behavior in service is a task of great complexity and need to understand and evaluate the current state that is on bridge and a more accurate mechanical characterization in failure. In recent years studies have been performed on structures to know the behavior under service loads. The aim was to characterize the behavior in service and, by extrapolation of the tests results to estimate the ultimate capacity of the bridge.

Davey (1953) and Chetoo and Henderson (1957) conducted tests on structures in apparent good condition. More recently, Boothby has carried out a test campaign on structures in service to take the measure of force through the passage of LVDT vehicles with different axle loads in key section of structure.

The load-deflection curves obtained indicated a linear elastic characteristic in the behavior of these structures under tested. The curve obtained in the five structures showed no plasticizers which could indicate the formation of hinges. Moreover, in comparison with tests carried to rupture, the graph remained linear up to about one third of the ultimate load, Boothby extrapolates the possible failure load of these structures is at least three times applied load. This point of view is debatable, since the curve, especially at its ending point, depends on the expected failure mode in the structure.




Figure 2-7 Service performance at the point of the point load applied (Boothby, 1995)

There are a number of tests which studies the global behavior of the infill inside the masonry arch bridges. Within these tests, a specific constituent dynamic identification method was used. These tests are performed using the impulse excitation technique, which investigates the interaction between structure and infill, and also the influence of the type of load applied with the dynamic properties of the structure.

2.3.2. Mechanism of collapse and theories

Although some theories relating to the thrust lines had been determined inaccurate, it was noticed that they are still an important concept. The thrust line will correspond with the location of a hinge. When the line of thrust becomes tangent to an alternate boundary, a hinge will develop at that location. It was determined that an arch needs a minimum of 4 hinges to create a mechanism. Some typical collapse mechanisms are shown in Figure 2-8.

Figure 2-8 Collapse mechanisms with a possible corresponding thrust line (Roca, 2008): (a) Typical 5-hinge collapse mechanism of an arch with symmetrical loading and geometry; (b) Typical 4-hinge

collapse mechanism of an arch with asymmetrical loading and/or geometry.




Using the ideas of collapse mechanisms and thrust lines, several ideas of identifying the capacities and limit states were proposed. The attention was directed towards finding the actual thrust line among the infinite solutions which can be drawn in a stable arch. The theory of the thrust line, however, was a tragicomedy of unsuccessful attempts to remove the static indeterminacy by means of empirical hypotheses or metaphysical principles.

Not until 1966 with Jacques Heyman’s work did an appropriate limit analysis appear. He introduced a formulation that was based on the plasticity theory rather than the inadequate elastic theory. His analysis is under the assumption of the following three hypotheses:

(1) Masonry has null tensile stress

(2) The compression strength of the material is infinite

(3) Sliding between stone blocks is impossible

Failure is due to the generation of a plastic mechanism.

Heyman stated an upper and lower (safe) bound theory within his hypotheses. The lower-bound theory states that if a thrust line can be found, for the complete arch, which is in equilibrium with the external loading (including self-weight), and which lies everywhere within the masonry of the arch ting, then the arch is safe (Heyman, 1966). The important part of this is that the thrust line need not be the actual thrust line. Thus, finding one satisfactory thrust line, it can be known that the arch cannot collapse and the need to examine failure modes is not required.

The upper bound theory states that under an assumed mechanism (arbitrarily providing sufficient number of hinges), equating the work of the external forces to zero will result in a load which is and upper-bound estimation of the actual ultimate load. The theory solves for the point at which the structure will fail, providing an upper approximation of its capacity.

Furthermore, the uniqueness theory provides an additional limit condition if a statically and cinematically admissible collapsing mechanism can be found. In other words, collapse will happen if a thrust line which causes the necessary number of plastic hinges to develop a mechanism. The resulting load is the true ultimate load with a corresponding ultimate mechanism and thrust line. In addition, a minimum and maximum thrust necessary for arch stability can be determined by positioning the reactions in the appropriate locations as seen in Figure 2-9.




Figure 2-9 Semicircular arch under self weight; (a) Minimum thrust and (b) Maximum thrust by

applying uniqueness theory. (Heyman, 1995)

As seen above, some of the ancient and early empirical criteria are still worthwhile for the assessment of masonry arches. Additionally, the geometrical rules can contribute to simple analysis such as to verify whether the structure was consistently designed according to contemporary criteria and gain a first and quick insight on the adequacy of the design and safety condition. Limit analysis depicts realistically the collapse and capacity of masonry arches. In combination with other tools, it can be a very reliable analysis.

Many additions to the theory have been studied and experimented; however this remains the fundamental principle behind them. More recently, numerical approximations and software analysis have been developed. As the focus of this paper is not the assessment of masonry arch bridges, these additions and more detailed descriptions of the above, will not be discussed. Appendix A provides a list of some available resources on the assessment of masonry arch bridges.




3. EXPERIMENT TEST, RESULT AND ANALYSIS

3.1 Construction of the bridge arch in the laboratory

The bridge is one of a series of arch bridges of masonry which was tested in the UPC in a program of experiments led by Prof. Pere Roca. This bridge was designed and built throughout the academic year 2000/01 as the first initial of two bridges, a lowered and segmental arch bridge. Its subsequent experiment at failure was made in early July 2001.

3.1.1. Geometric description of the bridge

The size of the bridge was made taking into account the constraints of laboratory space, availability and capacity of the test apparatus, mainly the loading frame, and of course control of budget, as already mentioned, is important in such experiments. This is a segmental bridge with masonry brick, whose main dimensions are as follows:

• Total length of bridge: 5.2 m. This dimension includes the length of the spandrel walls that extend one meter from the end of the arch.

• Arch length: 3.2 m. This is the actual length that saves the bridge arch.

• Height of arch center: 0.65 m.

• Thickness of Arch: 0.14 m. It is a constant and equal to the brick.

• Thickness of the infill in abutment: 0.8 m. This thickness refers to the height of infill out of the arch in the abutment.

• Height of the bridge: 1 m. This dimension is included the height of the slabs where the arch was built on. It is concrete slabs of 0.2 m plant height and 1 m rectangular side.

• Thickness of the spandrel wall: 0.14 m. Again it is the thickness of the bricks that make up the spandrel wall of the bridge.

• Width of the bridge: 1 m.

In Figure 3-1 you can view the dimensions represented in every part of the bridge:




Figure 3-1 Geometry of the bridge tested in the LTE of the UPC (dimension in m)

As shown in the figure 4-1, it is a pretty low arch. Spandrel wall lasted up to 1 m above where the arch starts to properly support the infill. It were placed two large concrete slabs to control displacements and transmit the arch’s force.

It was decided to have a metal frame on each side of the bridge that could absorb the horizontal forces transmitted to the abutments. These metal structures should also act directly as containment of structures and the infill lengthwise over the bridge. Finally it was decided to provide resistance linking them together by stiffening ties, so that the horizontal reactions remain absorbed by metal structure. Measuring forces in the bars, it could also record the horizontal reactions that occur during the experiment.

3.1.2 Materials used in construction

In this section we describe the materials used to build the bridge arch:

• The masonry used was composed of solid bricks of plan dimensions 13.5 × 28.5 cm2 and 4.5 cm thick, commonly known as 'totxo català'. We used about 500 bricks to build this arch ring and the spandrel walls. The compressive strength of test specimens (consisting of three bricks) were tested in the laboratory was 130 kp/cm2.

• The mortar used to make the joints between the bricks was an M80, and all joints were flush.

• It was used about 400 liters of concrete for slabs where local the support for arch and the spandrel walls. These slabs were armed with rods B 500 S, with an approximate weight of 40kg.




• The infill was loose sand commonly used in the manufacture of concrete. Its density is 1.55 ton/m3 and it was dry, Proctor density (which is the one trying to get through the compaction of fill inside the bridge) was 1.86 ton/m3. It is used to fill the bridge with this sand to the height of the top and it took about 2.5 m3. The friction angle of these sands compacted once took values between 35 º and 40 º.

• The two metal structures located on both abutments of the bridge were formed by a metal plate that became rigid by welding a vertical metal profiles. The number of these elements and their dimensions are:

- 2 metal sheets of 100×100 cm2 and 5 mm. thick.

- 5 vertical profiles of 5 cm square holes inside, 4 mm thick and 100 cm in length welded to each plate.

- Soldiers vertical profiles, 4 horizontal rails (each side) 140x80x6 mm rectangular holes of 118 cm in length, with holes in the ends for threading the bars.

- 8 bars of φ 25 at the ends to anchor horizontal rectangular profiles.

With these materials and these amounts, approximately total weight of the masonry bridge turns out to be about 7 tons.

3.1.3 Bridge construction process

Once determined and assigned the position of bridge structures in the laboratory and the hydraulic jacks used for its loading, it is then load on the slab. The bridge is situated so that the load can be applied on the position of quarter-span of the bridge, which is the area where the bridge was loaded at the most critical point.

The first step is to build reinforced concrete slabs on which the arches start. These slabs have a thickness of 1 m aside on the ground, and a height of 20 cm. They have an inclined plane perpendicular to the arch area of the vault so you can put the first slice of the brick and thus realize the “embedded”.

Once the concrete has hardened and has acquired a minimum strength, we continue with the construction of the masonry arch and the spandrel walls. For the arch, it was previously built a wooden form; the dimensions can be seen in Figure 3-2.




Figure 3-2 Formwork timber of the vault

The metal structure had two main functions. First is to contain the soil inside the space during the construction of the bridge. Secondly, the role that restrain slips of the slabs in abutment and resist the push force as a result of application of load and movement of the arch.

Figure

3-3 Elements and assembly of the metal structure of horizontal restraint

The infill used was composed of a kind of sand which is usually used in making concrete. Its density was 15.5 kN/m3 and dry. Proctor test was performed at the Laboratory of Normal Transport of the UPC for measuring optimum moisture of the infill (following the rules NLT-107/91). The results obtained optimum humidity was approximately 5% and a maximum density is 18.6 t/m3. The results obtained in the test and the curve Proctor moisture-density, are shown in Table 3-1.




Table 3-1 Results of Standard Proctor, Moisture-density curve

After that, the compaction process was made as shown in the following figures.

Figure 3-4 Layers of sand and the optimum moisture content before and after being compacted




Layers Left side Right side (where the load is applied) 1 1,92 1,93 2 1,98 1,96 3 2,01 1,98 4 1,94 2,05 5 1,89 1,90 6 1,93 1,98 7 2,02 1,99

Table 3-2 Densities measured in each layer

Layer 1 is the deepest and the layer 7 is the most superficial. The density measurement is of great importance not only for uniform compaction, but above all in view of the characterization of the infill to the analysis of results and their comparison with those predicted by numerical methods of analysis. Thus, from the density and moisture it can be obtained a good approximation of the elastic modulus E.

After the compaction process is complete, the surface of infill and the bridge can be considered finished and ready to be implemented and tested. The final aspect of the bridge reflected in Figure 3-5.

Figure 3-5 View of the bridge ready for testing

3.2 Implementation of the bridge

The instrumentation and monitoring the behavior of the bridge was subject to the availability of measuring instruments in the laboratory at the time of experiment. Compared to other studies, the degree of implementation was not high. This is because, as the initial experiment, the greatest




interest was in determining the collapse load and observing the failure mechanism. In this sense, the implementation of the experiment was sufficient.

Apparatus used as instrumentation during the test were:

- load cells to measure the force on the bars. There were only 4 of these cells. These 4 cells were placed in the 4 bars on the same side taking advantage of the symmetry of the bridge and load, although lacking of contrast and verification not measuring the forces on the opposite side bars.

- A load cell under the hydraulic jack cylinder to measure the load applied to the bridge.

- Extensometers to measure the arch in the process of three loading placed: one from the point of load application, one in the center of the arch ring and the third in the point symmetrical to the application of the load from the center ring.

- Strain gauge embedded in the infill of the bridge to measure ground motion, placed 4 of these devices at different depths and all in the same bracket of the bridge, closest to the point of load application as there would be greater distortion. In addition to this implementation, it took a few data processors to obtain the measurements obtained by instruments on computer.

Depths which were embedded these 4 gauges are:

- Gauge 1: 28 cm. deep




Figure 3-6 Strain gauge and ram




Figure 3-7 Extensometers placed under the arch

- Data Processors

These data processors, which in this case was a PC connected to a card that made the implementation measures in a computer file, were connecting to instrumentation after it was fully installed. Another device used was applied to the bridge arch. The device was connected to the hydraulic jack, which also was connected to the PC to always know the value of the applied load. In Figure 3-8 you can see an overview of PC and the device that made the rate of load application:

Figure 3-8 Data processors

3.3 Conduct of the experiment

The loading process is recorded by video and a large number of photographs were taken throughout the test to track the development of cracks.

3.3.1 Process of implementation of the load

The load was applied at a rate of 0.05 kPa, which is to say, given the size of the increment, to give 10 kN load takes a little over 4 minutes (the surface of load application device is 82 cm2). The evolution of cracks in the masonry during the experiment were recorded, and directly marked and numbered




on the bridge. The cracks are classified according to occur in the spandrel wall East or West side. They were marked black for the cracks developed before 5 tons of load and red for the cracks given between 5 and 10 tons of load. While observing and marking these cracks, people would take pictures of the cracks as they appeared and the experiment was recorded with a video camera. Below are the main cracks detected during the loading process.

When it moving to 7 tons, the movement of concrete slab is abruptly increased, accompanied by another rise of the concrete slab. The shape of these movements is shown in Figure 3-9. The movements of the concrete slabs made it possible two separate rotations of the arch start without forming plastic hinges. In other words, we can say that the arch was not tested fixed foundation. Therefore the 'free' rotation of the arch at these points could be given without the prior formation of a plastic hinge.

As a result, the formation of two plastic hinges, one under load point and one at a point close to the symmetry point of load point, was enough to transform the structure into a mechanism. Thus, we can say that the form of failure was caused by a 4 hinge mechanism, two of which, were cause by insufficient rotational restraint of the concrete slabs for supporting the arch.

Figure 3-9 Mechanism of rupture of the bridge tested (Tesina, 2001)

In reaching the load of 7.24 tons, it can be clearly seen that the shifts forming a plastic hinge. This caused large cracks under the load and one other point and cracks in the infill, and small detachments of material in slabs and under the dome. The loading process is stopped before these movements produce the final rupture of the bridge. All this is reflected in the figures that follow.

Figure 3-10 Full view of the 2 hinges from the east




Figure 3-11 Crack separation between the vault and the spandrel wall on west side

3.4 Analysis of test results

Data resulting from monitoring the behavior of the arch during the experiment were processed by the equipment described above and stored in a spreadsheet. As already explained the data were taken every 5 seconds in each of the instruments provided:

• The load cell, which provided the value of the force applied to the quarter-light position.

• The 4 strain gauges embedded at different depths along the height closest to the load applied, taking values of soil deformation in the area.

• The 3 motion transducers located in the middle and quarter ring under the extrados, which provided values of the deformation in the arch.

• 4 load cell measures the horizontal reaction force on the bars.

3.4.1 Data on the load applied

The rate of application of the load was constant over time and equal to 0.05 bar / second, which considering the surface of the device is about 1 ton every 4 minutes. We chose a rate of load application slow enough to observe the formation of cracks and photographed. Figure 3-12 shows the evolution of the load applied by the hydraulic jack at the time.




Figure 3-12 Evolution of load over time

As shown in Figure 3-12, the ultimate load of the bridge tested reached a value of 7.24 tons. This result is well below that predicted by numerical analysis methods (Soms, 2001), or the value that could be extrapolated from other similar experiments till now. Because as explained above, insufficient rotational restraint of the concrete slabs on supporting part of the arch, result in lifting such a slabs close to collapse. This means that there was a shift in the arch sections started without the need of forming plastic hinges at these points. Thus, the formation of a mechanism was possible without the all the necessary 4 hinges fulfilled.

3.4.2 Results obtained from gauges embedded in the infill.

The 4 gauges were placed as close as possible to the footpoint of the load side of the bridge. The arrangement of the depth of gauge was:

• Gauge 1: 28 cm. deep




The arrangement of the gauges intended to follow approximately the law of real passive resistance in other experiments to break, in which the push force grows to a certain depth and then decreases because of friction experienced by the infill on the bottom surface that as shown in Figure 3-13




Figure 3-13 Difference between theoretical and experimental passive resistance

The third gauge was placed at the depth where is approximate the maximum passive resistance, and the lowest gauge should represent the decrease in horizontal resistance due to friction between infill and lower surface that contains it. However, as shown in the figures below, the third gauge readings were not correct and, these were possibly the most important gauges, representing the thrust obtained very accurately.

Although what the gauge measures are deformations, a simple product of them by the gauge modulus (10,000 kp/cm2) results in pressure increase that the infill spread to the gauge. The following figures show the strain of that with the evolution of the load:

Figure 3-14 Deformations of the upper gauge (Gauge 1)




Figure 3-15 Deformations of Gauge 2

Figure 3-16 Deformations of Gauge 3

Figure 3-17 Deformations of the lowest gauge (Gauge 4)




The first thing to note is that the gauge 3 readings are not valid. This may be due to unwillingness of the device or to a poor state of the gauge, but it is probably the latter because of errors, since, the placement of the gauges, carefully made, was not of much difficulty.

The figures show how the upper part of the infill, from gauge 1 and 2, did not begin to deform until it reaches the 4 tons of load. This period coincides with the appearance of major cracks in the arch area under the load, which can be considered the formation of a first plastic hinge behavior and neglect of elastic deformation of the arch apex.

The behavior of the bottom of the infill has been proved to be very different. As shown in the figures, gauge 4 was in this area and the deformation rate is fairly constant, starting from the early stages of loading.

In Figure 3-18 shows the evolution of the deformation of the infill over time, trying to represent the law of passive resistance, not being able to use the gauge data from third gauge. No clear trend is expected in this passive resistance. Probably the readings of third strain gauge should be higher than those of lower gauge, which would prove the decrease in thrust caused by the friction with the bottom surface that contains it. The failure of the device made it impossible to check this prediction.

Figure 3-18 Evolution of deformations in depth over time




It then showed the results of motion transducers placed at different points of the arch. These gauges were located at the half and quarter points of ring, measuring vertical displacements. Note that the arch also suffers horizontal movements which were not recorded. These, in any case, are smaller and of less interesting. The figures below show the evolution of the displacement at these points with the load.

Figure 3-19 Displacement at the point of load application

Figure 3-20 Displacement in the center of ring




Figure 3-21 Displacement at point symmetrical to the load application point

The loading point experienced a decline, while the points in the symmetry of loading point rose as expected. The displacement at the time of failure proved to be very similar in the three points, approximately 15 mm, or what is the same as 1 / 230 of the ring. After the formation of the movement mechanism, especially the lifting of the concrete abutment of the arch, the extensometer was finished its mission.

As can be seen in the three figures, the displacement grows with approximately constant speed to 3 tons of load. This period coincides with the appearance of major cracks in the arch area under the load, which can be considered the formation of a first plastic hinge behavior and neglect of elastic deformation of the arch apex.

So there was, until now, a significant elastic deformation of the arch with a displacement under the load of around 3 mm.

After the change of slope with the formation of the first hinge under load, the deformation happened to grow more quickly to where may be the formation of the second hinge, just reached the 7 tons of load. At this point there is another change of slope increasing again on rate of deformation. These two slope changes seem to coincide with the formation of plastic hinges.

In the plots of load-displacement results of the experiment in the technology laboratory of the UPC, structures can be seen as the collapse was reached almost immediately after the formation of the second hinge. As explained above, insufficient rotational restraint of the concrete slab abutments on supporting part of the arch, result in lifting such a slabs close to collapse. This is to say that there was a displacement in the arch sections starting without the need of forming plastic hinges at these points.




In any case, so far the evolution of deformation with loading on the arch was as expected, beginning in the elastic range and increasing the strain rate after the load with appearance of plastic hinges.

3.4.4 Results obtained with the load cells.

Here is the data obtained from load cells located in the stiffening bars. Figure 3-22, on the other hand, shows the evolution of the load in the different bars as load increases.

Figure 3-22 Evolution of the load on the stiffening bars

It can be seen that the trend of the 4 graphs is very similar. It gives first a slow load, until a change of slope and the load on the bar accelerates increased to break the bridge.

The evolution of the load on the bottom bar would show a more constant increase in the horizontal reaction that generates in the arch. At least to situations close to collapse, when the magnitude of the response grows more quickly. The total horizontal reaction at each abutment at the time of collapse exceeded 11 ton.

The initial value of each graph represents the load that was given to each of the bars before beginning the test, to confine the land in the abutment.

The upper cell is with the smallest fore increases detected since it is located in the most favorable position, which holds the infill push on it. For this reason the force only increases to 400 kp approximately.




The second and third cells show higher force increases as they contain the area filled with higher thrusts, taking force increases close to 800 kp in both cases.

The lowest cell detects a load increase of about 710 kp. These bars were below the height of the slabs out of the foot in order to catch any horizontal movement. It is the fact that the bottom bars are to be given greater initial load before starting the loading process. This cell also experiences a decline of about 1 ton load when removing the load applied to the bridge.

3.5 Analytical prediction using RING 1.5

3.5.1 General information

As a second step of the research, the experimental results will be used to asses the ability of different calculation methods to predict the general behavior and the ultimate capacity of masonry arch. For that purpose, the limit analysis software RING 1.5 is used to estimate a lower bound of the ultimate load and the collapse mechanism of the masonry arch bridge.

In the RING 1.5 model the lateral confinement of the un-cohesive infill has been neglected because, given its relative small volume, it is support to have very little effect on the response the bridge tested, thus, only the stabilizing effect of the infill weight has been considered. The structure consists of zero tensile strength joints. To be simplified, the effect of spandrel wall is neglected. The compression strength of the masonry is accounted for as a reduction of the available depth of the structural compounds. The maximum shear forces are limited by the Mohr-Coulomb criterion using the values of the cohesion and the angle of friction provided in table 3-3. The load is applied on the surface of the infill with a width of 200 mm at 1/4 of the ring. The horizontal pressure of soil is set up to zero in this case.

Figure 3-23 Geometrical definition of RING model




Geometry (all distances in mm, angles in radians)

Global: Bridge width Infill depth 1000 950

Abutment: Abut Height Width(top) Width(base) No of blocks 170 1000 1000 2

Span: Shape Span Rise Auto angle LHS angle RHS angleSegmental 3200 650 Yes 0.799 0.799 Ring No. of Blocks Thickness

1 56 140

Material Properties (unit weights in kN/m3, stresses in N/mm2, angles in radians)

Masonry: Unit Weight Radial friction Tangential friction Crushing Strength 18 6 0.5 14

Convergence tolerance (%) 0.5

Infill: Unit weight Angle of friction Cohesion Horizontal pressure 18 38° 0 None

Load disperse angle Load disperse type 0 uniform

Load Cases (all distances in mm)

No. 1 Vehicle Position Width LC1 800 200

Table 3-3 Model information using RING 1.5

3.5.2 Results and analysis

After the geometrical and material properties definition, the calculation had been made to obtain the analytical result.

Analysis result

Critical load factor = .24 (load case 1)

Ultimate load = 60×0.24 = 14.4 kN

Critical collapse mechanism:




Figure 3-24 Collapse mechanism of masonry arch bridge model

As shown above, the ultimate load reached by software is much lower than what it is obtained in the experiment campaign. To explain this, we need to consider the assumption adapted in this stimulation. Firstly the horizontal confinement is ignored here, which might act as an important factor influencing the infill and masonry behavior (it will be discussed in the next chapter). Secondly, horizontal pressure is neglected; only consider the vertical contribution of infill weight. Last and foremost, the effect of spandrel walls is not considered in this experiment. The confinement and support function of spandrel wall is obviously clear from the previous research; modeling without spandrel walls caused a lower ultimate load than real experiment is acceptable.




4. FINITE ELEMENT ANALYSIS OF PLANE STRESS MODEL FOR MASONRY BRIDGE

4.1 Introduction

Following the analytical study of the bridge, a finite element analysis is made to stimulate the experiment. In order to begin from the simpler one and know clearly how the finite element model works, a 2D model is first made and then, in the next chapter, 3D model had been built to produce the stimulation. The finite element method software iDiana and Diana is used in this research work. Diana is an extensive multi-purpose finite element software package that is dedicated, but not exclusive, to a wide range of problems arising in Civil engineering including structural, geotechnical, tunnelling and earthquake disciplines and oil & gas engineering. DIANA is a proven and tested software package; the program's robust functionality includes extensive material, element and procedure libraries based on advanced database techniques, linear and non-linear capabilities, full 2D and 3D modeling features and tools for CAD interoperability. With the continuous demand for more efficient utilization of resources and materials, together with the increasingly complex nature of engineering structures, Diana is the all in one solution, which provides the competitive advantage when tackling design and assessment work in a huge range of civil and geotechnical scenarios. According to that, it can subsequently be concluded that Diana is an adequate software to cover this modeling work.

The procedure used for developing the models was as follows:

- Develop the form of the structure in AutoCad.

- Import the drawing as .dxf format to iDiana.

- Define units and plane stress approach.

- Define mesh type and assign to surfaces.

- Define sets of surfaces and lines that relate to specific materials or constraints.

- Set up constraints.

- Define materials and assign to the relevant surfaces.

- Define gravity loading, pressure loading and load case.

- Glue the gap of interface and make meshing.




As the first step of finite element analysis, the 2D modeling approach made some simplifications, which is to better understand and compare with the analytical solution and get ready for the 3D model construction. The 2D model had been setup to plane stress model, according to which the element types are chose later. In the 2D model, the effect of spandrel walls are neglected, only the infill is put on the arch. The concrete slab abutments are located at the bottom of the infill and the foot of the arch to anchor the whole structure. Two steel plates were placed at the lateral sides of the infill and connected by steel ties. And the soil infill part had been divided into three layers with different Young’s modulus to realize the stimulation that the density and elasticity modulus varies along the depth. And interface element was introduced here to make it possible the detachment between the infill and masonry arch, which is more close to real situation.

4.2 Geometry definition

The geometrical size is made with the design of experimental bridge. The units are setup as millimeter for length, kilogram for weight and Newton for force. The width of the structure is 1000 mm. In finite element analysis, load directly on the infill, which is a soft material, may cause too much stress concentration, subsequent diverge for nonlinear analysis. So a concrete bean is design on the loading point with a width of 200 mm to disperse the stress.

As mentioned before, in this part, to better simulate the real boundary conditions of the masonry bridge experiment, two steel plates are placed at the two lateral sides of the infill and connected by four steel ties, which supply the lateral constrain capacity during the loading procedure. The steel plate thickness is 1 cm. In the experimental campaign there were eight ties, four in each side. However, to be simplified only four ties were made and the cross section area of the tie is 981.25 mm2, corresponding to two ties place at the same height.

Three layers of soil infill are construct here, the depth of each is 273, 290 and 300 mm respectively from top to the bottom. The gap for interface element between infill and masonry ring is 50 mm.




Figure 4-1 Geometrical definition of plane stress finite element model

4.3 Element type

As describe above, choose different type of element for different component is necessary. In this part of research, three types of element are made use of: plane stress element, truss element and interface element. With triangle or quadrilateral layout, the plane stress element is assigned to the surfaces belong to soil, masonry, concrete and steel plate. And truss element is used to stimulate the steel ties, to supply lateral constrain to the infill. The interface element is placed between the soil infill and the masonry arch as well as the concrete slab abutment.

4.3.1 Plane stress element

Flat plane stress elements must fulfill the following conditions with respect to shape and loading (Figure 4-2).

Figure 4-2 Plane stress elements characteristics




They must be plane, i.e., the coordinates of the element nodes must be in one flat plane, the xy plane of the element. They must be thin, i.e., the thickness t must be small in relation to the dimensions b in the plane of the element. Loading F must act in the plane of the element.

Element CQ16M – quadrilateral, 8 nodes

Figure 4-3 CQ16M element

The CQ16M element (Figure 4-3) is an eight-node quadrilateral isoparametric plane stress element. It is based on quadratic interpolation and Gauss integration.

Element CT12M – triangle, 6 nodes

Figure 4-4 CT12M element

The CT12M element (Figure 4-4) is a six-node triangular isoparametric plane stress element. It is based on quadratic interpolation and area integration. Typically, this polynomial yields an approximately linear strain variation in x and y direction. By default Diana applies 3-point integration.




4.3.2 Interface element

The structural interface elements describe the interface behavior in terms of a relation between the normal and shear tractions and the normal and shear relative displacements across the interface. Typical applications for structural interface elements are elastic bedding, nonlinear-elastic bedding (for instance no-tension bedding), discrete cracking, bond-slip along reinforcement, friction between surfaces, joints in rock, masonry etc.

Element L8IF – line, 2+2 nodes, 2-D

Figure 4-5 L8IF element

The L8IF element is an interface element between two lines in a two-dimensional configuration (Figure 4-5). The local xy axes for the displacements are evaluated in the first node with x from node 1 to node 2. Variables are oriented in the xy axes. The element is based on linear interpolation.

4.3.3 Truss element

Truss elements are bars which must fulfill the condition that the dimensions d perpendicular to the bar axis are small in relation to the bar’s length l (Figure 4-6). The deformation of truss elements can only be the axial elongation.

Figure 4-6 Truss elements




Element L2TRU, straight, 2 nodes

Figure 4-7 L2TRU element

The L2TRU element (Figure 4-7) is a two-node directly integrated (1-point) truss element which may be used in one-, two-, and three-dimensional models. This polynomial yields a strain εxx which is constant along the bar axis.

4.4 Material properties

The primary challenge of this finite element method model is to design proper soil and masonry properties aiming at stimulate the real environment of the experiment. As we know, soil is a kind of loose material, the choice of proper elasticity modulus is crucial for soil infill behavior. As described before, the compaction processes influence the soil infill properties very much, making different densities for different layers from bottom to top. With the measurement of densities and moisture, it can be obtained a good approximation of the elasticity modulus E. Getting a approach that elasticity modulus E varies from top to bottom, it is used a formulation as follows:

0.32E Z= i

Z is the depth of the soil from top surface. Units used are cm for Z and Mpa for E. Close to surface the elasticity modulus is considered to be almost zero.

To realize this situation in iDiana, I simplified the soil infill into three layers, shown in Figure 4-8 in different colors.




Figure 4-8 Three soil layers of Masonry Bridge

The Mohr-Coulomb plasticity model was used to describe the soil. The material properties of soil are in Table 4-1.:

Linear properties Nonlinear properties

Depth (mm)

E modulus (Mpa)

Poison ratio

Density (Kg/mm3)

Cohesion Friction angle

Dilatancy angle

Soil A 273 4.352 0.2 1.84E-6 0.02 38° 38°

Soil B 290 13.38 0.2 1.84E-6 0.02 38° 38°

Soil C 300 22.82 0.2 1.84E-6 0.02 38° 38°

Table 4-1 soil properties for finite element model

As for masonry material, it is to know that this part is of great importance. The masonry arch is the major bearing structure, which once collapse the whole structure will loose function. The energy based total strain fixed crack model was used to stimulate the masonry. The tensile behavior includes a linear softening while the compression curve is of parabolic shape. The masonry properties used are as follows:




Masonry properties

Total strain fixed crack

linear softening in tension

Constant shear retention

Parabolic diagram in compr.

Elasticity modulus (Mpa) 5120

Poison ratio 0.18

Density (Kg/mm3) 1.8E-6

Tensile strength (N/mm2) 0.05

Mode-I tensile fracture energy 0.03

Shear retention factor 0.01

Compressive strength (N/mm2) 21

Compressive fracture energy 5.0

Table 4-2 Masonry properties

In this research work, concrete and steel were working within elastic range. So only linear material properties was assign to them.

Concrete Steel

Elasticity modulus (Mpa) 34000 2.0E5 Poison ratio 0.15 0.27

Density (Kg/mm3) 2.2E-6 7.85E-6

Table 4-3 Concrete and steel properties

During the developing of the 2D finite element model, interface is placed between masonry arch and soil. However, for the convenience of construction, the interface is placed also setup between soil infill and concrete slab abutment, but the linear normal stiffness is high to ensure that the relative movement of them is restrained. Simple frictional contact surface for the interface between soil and arch with no hardening has been assumed. The material properties of interface element can be seen in the table below.




Interface

between soil and arch Interface

between soil and concrete

Linear normal stiffness 1.0E+6 1.0E+8

Linear tangential stiffness 1.0E+6 1.0E+8

Density 1.0E-20 1.0E-20

Cohesion 0.05 Friction angle 38°

Table 4-4 Interface properties

4.5 Meshing, Boundary conditions and Load

As mentioned before, for soil, masonry and concrete, plane stress element is chose. Shown in Figure 4-9, the quadrilateral 8 nodes element CQ16M is used for rectangle surfaces and triangle 6 nodes element CT12M is used for triangle surfaces. In this Figure also can see the meshing divisions. Because one purpose of this trial is to stimulate the ultimate capacity of the masonry arch, when the masonry cracks, the smaller element size it is, the more accurate the result is. According to that, the element size is made. To be emphasis, the 8 nodes quadratic interpolation element CQ16M is made use of instead of 4 nodes linear interpolation element Q8MEM to get fine result when it come to nonlinear cracking.

Figure 4-9 Meshing divisions of structure




Figure 4-10 Meshing detail on the masonry arch

In the previous chapter it can be found that during the experiment, the concrete slab can be considered as fixed on the ground, the lateral supporting of soil and spandrel walls are supplied by the metal structure which is consist of steel plate construction and 8 steel ties, 4 on each side. So in the 2D model, the bottom surfaces of the concrete slab are fixed in Y direction and free in other direction.

Figure 4-11 Boundary condition on the 2D model




Self-weight and pressure on the loading concrete beam is applied on the structure. The pressure is 70 N/mm. Considering the plane stress model thickness 1000 mm and the length of the concrete beam 200 mm, the total pressure load applied is 14 kN.

Figure 4-12 Self-weight of 2D model

Figure 4-13 Pressure on 2D model




4.6 Results and analysis

The analysis processes are ran in Diana program. First step is linear analysis and then nonlinear approach is made for the calculation. As for the nonlinear analysis, steps for loadcase1 self-weight is 0.5(20) while for loadcase2 pressure load is 0.005(300), maximum number of iterations is 50, convergence tolerance for displacement, force and energy are 0.01, 0.01 and 0.0001 respectively. The iterative method used is Regular Newton Method. Arc length control is developed in this model aiming at get more complete graph of load-displacement. The ultimate load factor obtained is 0.918, which means the ultimate load is 12.85 kN. This result is close to the analytical solution, which is 14.4 kN. In the following parts show the results from linear and nonlinear analysis.

4.6.1 Linear analysis results

The properties used for linear analysis are mentioned before. The load applied on the structure is 14 kN.

Figure 4-14 Deformation shape of the structure and the arch

In Figure 4-14, the maximum displacement located in the part under the loading beam. The beam was punching against the infill. On the arch, the part under the loading point was descending and the part symmetric to it is rising as expect.




Figure 4-15 Maximum principle stress

It is seen in Figure 4-15 that the principle stress is concentrated in four areas on the arch where the hinges were expected to develop. There was also stress concentrated on the edge of the contact between loading beam and soil infill.

The linear analysis only supplies the preliminary results regarding the globe deformation shape and stress distribution. In order to be aware of the ultimate load as well as collapse mechanism, the nonlinear analysis is inevitable to be performed. The result is shown in the next part.

4.6.2 Nonlinear analysis results

In this part, firstly this model is undertaken the nonlinear analysis and after that, some other models with different boundary conditions or material properties are carried out for the purpose of comparison, which will be discussed later. The nonlinear material properties are described in the part 4.4. Both masonry nonlinear and soil nonlinear properties are utilized. The concrete abutment and loading beam as well as steel bracing system are under linear range.




Figure 4-16 Deformation on the 2D model

Figure 4-17 Deformation on the arch

It can be seen in Figure 4-16 and Figure 4-17 the deformation is quite reasonable. The maximum displacement is on the loading point under the concrete loading beam where the beam impact into the infill. The area under loading point is descending and the part symmetric to it is rising. The maximum deformation is 3.24 mm on the concrete beam and 0.43 mm on the arch. The deformation




on arch is consistent with the result obtained from the experiment that probably two hinges were developed on the upper part arch, making the structure able to rotate like that.

Figure 4-18 Principle stress on the bridge

Figure 4-19 Principle stress on the arch




As shown in Figure 4-18 and Figure 4-19 the maximum principle stress is 0.59 N/mm2. The principle stress is extremely high at four places. As we know, the masonry arch collapses only when four hinges mechanism has been developed. In Figure 4-17, the parts with red color is where the stress exceed the tensile strength 0.05 N/mm2, which means these parts had cracked, plastic hinges were formed in that places. Consequently, we can say that the 2D finite element model did a good prediction on the collapse mechanism of the masonry bridge.

Figure 4-20 Deformation on the arch under the loading point with load increment




Figure 4-21 Deformation on the arch at place symmetrical to loading point with load increment

Figure 4-20 and Figure 4-21 show the load – displacement graph at node on intrados of the arch under loading point and the symmetrical point of it. The graph corresponding to node under loading point is descending, while the one for node symmetrical to point under loading point is rising. These features are consist with the globe deformation. It can be seen that before about 0.4 times the ultimate load, both graph behavior a linear characteristic; after that, nonlinear behavior had been developed.

(a) (b)




(c) (d)

(e) (f)

Figure 4-22 Stress evolution with loading process on masonry arch

The loading processes are as (a) 0 kN (b) 2.45 kN (c) 4.2 kN (d) 9.1 kN (e) 11.886 kN (f) 12.852 kN

From the evolution process it is easy to say the first hinge developed at the left foot of the ring. After the formation the first hinge, as the load increasing to about one third of the ultimate load, the second plastic hinge had been observed on the arch under the loading point. When reaching the load of 9.1 kN, the third hinge on the point symmetrical to loading point was found beginning to appear. And close to the ultimate load the last hinge formed to make the structure a mechanism. This is the finial collapse mechanism for the masonry arch bridge. According to the finite element analysis results, it offered a prediction for the order of the hinges formation.




4.6.3 Comparison with finite element model of different masonry tensile strength

(a) (b)

(c)

Figure 4-23 principle stress of model with 0.02 N/mm2 masonry tensile strength (a), 0.05 N/mm2 (b) and 0.08 N/mm2 (c)

For the purpose of comparison, the models with less and more masonry tensile strength are made. Compared to the previous model with 0.05 N/mm2 masonry tensile strength, the new models adopt parameters of 0.02 N/mm2 and 0.08 N/mm2. The principle stresses of the three models are shown in Figure 4.21. From Figure 4-23 it can be seen that the four hinges developed most obviously in the 0.02 N/mm2 tensile strength model and became less and less obviously with the increase of masonry tensile strength. This is because reducing the tensile strength of masonry makes the structure softer and deform more than the model with a stiffer masonry. So the cracking can be developed in a large area than others.




0

2

4

6

8

10

12

14

-0.450 -0.400 -0.350 -0.300 -0.250 -0.200 -0.150 -0.100 -0.050 0.000

Deformation (mm)

Load (kN)

0.05 tensile strength 0.02 tensile strength 0.08 tensile strength

Figure 4-24 Load-vertical displacement comparison between models with 0.02, 0.05 and 0.08 N/mm2 masonry tensile strength at one node on intrados of the arch under the loading point

The Figure 4-24 shows the load-vertical displacement graph for models with 0.02, 0.05 and 0.08 N/mm2 masonry tensile strength respectively. This graph is acquired from the node on intrados of arch under the loading point.

As shown in figure 4-24, the deformation of model with 0.08 N/mm2 tensile strength is smallest in each step, the model with 0.02 N/mm2 tensile strength perform most deformation in each load step. The less masonry tensile strength makes the structure less stiff and able to undertake more displacement.

4.6.4 Comparison with finite element model of different soil layers

It is easy to make a model with uniform soil properties base on the previous model. Only change the soil properties can apply this. In the model with only one soil layer, all the soil material properties from different layers are setup as follows:

Young’s modulus: 37 Mpa

Poison ratio: 0.2

Density: 1.84e-6 kg/mm3




Cohesion: 0.02 N/mm2

Friction angle: 38°

After application of the parameters, nonlinear analysis was performed to obtain following results.

0

2

4

6

8

10

12

14

16

18

-0.400 -0.350 -0.300 -0.250 -0.200 -0.150 -0.100 -0.050 0.000

Deformation (mm)

Load (kN)

one soil layer three soil layers

Figure 4-25 Load-vertical displacement comparison between model with one soil layer and three soil layers at one node on intrados of the arch under the loading point

Figure 4-25 shows the graph about load-vertical displacement concerning to model with only one soil layer and model with three soil layers. The graph is played at the node on the intrados of arch under loading point.

During the experiment, the soil is compacted by layers; this treatment as well as the loose nature of soil material makes the elastic modulus different along the depth. So make different soil layers is a closer stimulation to the experiment. From Figure 4-25 it is seen that the vertical displacement is smaller for model with only one soil layer in every step. And the ultimate load obtained for model with one soil layer is higher than from the model with three soil layers. As in the one layer model, a higher elastic modulus is used than the three layers model, so the soil infill is stronger and has more capacity to take load. As a result, the masonry arch take less load and perform less deformation at the same loading level with the model with three soil layers. Since the masonry arch capacity is same in the two models, so the model with a stronger soil infill (one layer model) could reach higher ultimate load is understandable.




4.6.5 Comparison with finite element model without steel bracing system

In order to know the function of steel bracing system and influence to the structural behavior, a model without steel ties and plates was constructed. In this model, with introducing the steel bracing system, the lateral sides of soil infill were directly fixed in X axis, the Y direction is free to move (Figure 4-26). The rest are the same as the previous model.

Figure 4-26 lateral restrain of soil infill

Figure 4-27 Comparison of maximum displacement on arch

In Figure 4-27 shows the comparison of maximum displacement for model with steel bracing and model without it. It can be seen the maximum displacement happened under the loading point; the value is 0.395 mm for model without steel bracing and 0.43 mm for model with steel bracing. It can




be conclude from it that the bracing system made the structure softer than fixing the lateral side of soil in x axis. This is a more real condition close to the experiment.

Figure 4-28 Principle stress on arch of model without steel ties

From Figure 4-28, only three hinges were developed, the one close to right foot was missing. This is because the X axis displacement of the infill was constrained; the stress was dispersed to the boundary and other places so the structure is less soft, it is too late to form the last hinge before ultimate load was reached.




0

2

4

6

8

10

12

14

16

18

-0.400 -0.350 -0.300 -0.250 -0.200 -0.150 -0.100 -0.050 0.000

Deformation (mm)

Load (kN)

without steel bracing

with steel bracing

Figure 4-29 Load-vertical displacement comparison between model with steel bracing and without steel bracing at one node on intrados of the arch under the loading point

I picked up the node on the intrados arch under the loading point to draw the load-vertical displacement graph and make comparison between structure with steel bracing system and the one without them. From Figure 4-29 we can see in the same graph the curves load-vertical displacement corresponding to the comparison. The inclination of the curve regarding the model without steel bracing appears to be sharper than the other. The displacement of each load step is smaller. From Figure 4-29 it can be said that using steel bracing system to constrain the infill make the masonry bridge less stiff than using the method that fix x displacement directly on the lateral boundaries. The steel bracing system enable the structure to deform more until the ultimate load is reached. According to that, as mentioned before, the collapse mechanism improved when using more real boundary conditions. So in this comparison it is found that adopting the boundary condition more close to the real experiment helps to stimulate the structure behavior and perform collapse mechanism.




5. FINITE ELEMENT ANALYSIS OF THREE DIMENSIONAL MODEL FOR MASONRY ARCH BRIDGE

5.1 Introduction

After the limit analysis and plane stress finite element analysis, the final step is to develop a three dimensional finite element model of the bridge. The erection of this model is very complex and time consuming, due to time limitation of this dissertation, the checking and calibrating are the most difficulties to be faced here. The definition of the three dimensional model is based on the plane stress model.

5.2 Basic assumptions

The geometry has been obtained from the description of the construction of the bridge. The three dimensional model had been built according to that. However, as the three dimensional analysis is quiet time-consuming, we decided to model half of the bridge based on symmetric characteristic of the structure. In this three dimensional model, symmetric constrain had been applied on the cutting section, the concrete abutment is fixed in all directions; lateral support is supplied by steel bracing system. The load is applied at 1/4 of the arch span only on the surface of soil by a concrete loading beam of 200mm×200mm×360mm.

Friction interface elements were introduced between the masonry arch and soil infill as well as between the spandrel walls and soil infill. The soil infill had been divided into three layers with different Young’s modulus (increase from upper layer to lower layer) to stimulate the real nature of the infill during the experiment.

5.3 Modeling process

The first step of modeling is to define the geometry of the structure. As mentioned before, in this dissertation, half of the bridge cutting along the longitude direction was made. The thickness of the spandrel wall was setup to 140 mm which is the same width of a masonry brick, and width of soil infill was 360 mm. The depth of the three soil infill layers from top to bottom were 273, 290 and 300 mm respectively. The properties of each infill layer will be discussed later in this part.

In Figure 5-1 shows the geometrical definition of the three dimensional model. In this model, the units used were mm for length, Newton for force and Kg for weight.




Figure 5-1 The geometry definition of the model

In case there exist three prism and four prism shape of bodies, two types of solid elements and two types of interface elements were chose to fit them. The meshing division was made smaller on the arch to obtain better accuracy at the place close to where the hinges form. The meshing configuration is shown is Figure 5-2.




Figure 5-2 Meshing configuration (from 45°view and front view)

Material properties used in this 3D model were mostly the same used in plane stress model. But some modification had been made in interface properties to get a better result. As used before, the stiffness of 1e+6 is too high for the interface between masonry arch and soil infill in this three dimensional model. That interface caused considerable tensile stress on the intrados of arch under loading point, which is not expectable. So trials had been made to reduce to stiffness of interface. After that, we found the tensile stress on the intrados of arch had disappeared. So we can say too stiff interface would cause too much tension on intrados of arch because movement between arch and infill is constrained. From the other point of view, the interface stimulation make the three dimensional model more realistic and more accurate by considering the relative movement of masonry arch and soil infill as well as spandrel wall and soil infill.

Regarding the elements used in the model, it would be mentioned that total five types of elements had been utilized:

TP18L – wedge, 6 nodes solid element




Figure 5-3 TP18L

The TP18L element (Figure 5-3] is a six-node isoparametric solid wedge element. It is based on linear area interpolation in the triangular domain and a linear isoparametric interpolation in the Z direction.

HX24L – brick, 8 nodes solid element

Figure 5-4 HX24L

The HX24L element (Figure 5-4) is an eight-node isoparametric solid brick element. It is based on linear interpolation and Gauss integration.

T18IF – plane triangle, 3+3 nodes, 3-D interface element

Figure 5-5 T18IF

The T18IF element is an interface element between two planes in a three-dimensional configuration (Figure 5-5). The local xyz axes for the displacements are evaluated in the first node with x from node 1 to node 2 and z perpendicular to the plane.




Q24IF – plane quadrilateral, 4+4 nodes, 3-D interface element

Figure 5-6 Q24IF

The Q24IF element is an interface element between two planes in a three-dimensional configuration (Figure 5-6). The local xyz axes for the displacements are evaluated in the first node with x from node 1 to node 2 and z perpendicular to the plane.

L2TRU, straight, 2 nodes

Figure 5-7 L2TRU

The L2TRU element (Figure 5-7) is a two-node directly integrated (1-point) truss element which may be used in one-, two-, and three-dimensional models.

5.4 Linear analysis

The material properties used for the linear analysis are summarized in the Table 5-1.




Properties Concrete MasonrySoil

(Layer 1)Soil

(Layer 2)Soil

(Layer 3) Interface Steel

Young’s Modulus (Mpa) 34000 5120 4.352 13.38 22.82 - 2.0E5

Poison’s ratio 0.15 0.18 0.2 0.2 0.2 - 0.27

Mass density (Kg/mm3) 2.2E-6 1.8E-6 1.84E-6 1.84E-6 1.84E-6 - 7.85E-6

Linear normal stiffness (N/mm2)

- - - - - 5 -

Linear tangential stiffness (N/mm2)

- - - - - 5 -

Table 5-1 Material properties for linear analysis

Based on the material properties above, the linear analysis process had been carried out. To be mentioned here, the load applied on the concrete loading beam was half the value of ultimate load, which means 7 kN load had been applied on the beam in this three dimensional finite element model.

Figure 5-8 Deformation of the masonry arch bridge




Figure 5-8 shows the deformation of the three dimensional model. It can be seen that the maximum displacement happened on the area close to concrete loading beam. The maximum displacement of that is 2.93 mm, which is the same with the plane stress model’s analysis result.

Figure 5-9 Deformation of masonry arch

In Figure 5-9 shows the deformation on the arch of the three dimensional model. The part that under the loading point is descending and the part symmetric to loading point is rising. This pheromone is consisting with the plane stress analysis result. The maximum displacement is under the loading point, the value is 0.332 mm.

Figure 5-10 Maximum principle stress of masonry arch bridge




Figure 5-11 Maximum principle stress on the arch

In Figure 5-10 and 5-11 shows the maximum principle stress on the masonry arch bridge and on the arch of it. It can be seen that four parts of the arch is under high tension. As mentioned before, the collapse mechanism of four hinges is the critical state for the failure of the structure, which could probably be developed in the nonlinear analysis later.

Figure 5-12 Minimum principle stress on the arch




5.5 Nonlinear analysis

After the linear analysis, the next step is to perform nonlinear analysis with the three dimensional finite element model.

The nonlinear material properties used are summarized in the tables below. The concrete material and steel were setup to work within linear range.

Nonlinear properties Cohesion Friction angle Dilatancy angle

Soil A 0.02 38° 38°

Soil B 0.02 38° 38°

Soil C 0.02 38° 38° Interface between soil and arch 0.05 38° 38°

Table 5-2 Nonlinear properties of soil and interface

Masonry nonlinear properties

Total strain fixed crack

linear softening in tension

Constant shear retention

Parabolic diagram in compr.

Tensile strength (N/mm2) 0.05

Mode-I tensile fracture energy 0.03

Shear retention factor 0.01

Compressive strength (N/mm2) 21

Compressive fracture energy 5.0

Table 5-3 Nonlinear properties of masonry

After the definition of nonlinear material properties, the nonlinear analysis was carried out. The load steps had been setup to automatic load steps. The maximum size was 0.01, minimum size was 0.0001. The ultimate load reached was 25.06 kN. Considering the model was a half bridge, for a complete bridge model, the ultimate load could be 50.12 kN, which is smaller than the experimental result.




Figure 5-13 Deformation of the three dimensional model

Figure 5-14 Deformation on the arch

In the Figures above show the deformation of the structure. As the similar, in Figure 5-13, the maximum displacement occurred under the loading beam. The maximum displacement is 9.78 mm. In Figure 5-14, the deformation of arch is consistent with the linear analysis result. The maximum displacement was located at the extrados of arch under the loading point, which is 0.947 mm. The part symmetric to the loading point is rising. In chapter 4, it is obtained in plane stress model the




maximum displacement on the arch was 0.43 mm, considering it is reached under the load of 12.85 kN. A in the three dimensional model, when the load reached 12.8 kN, the displacement on the arch under the loading point was 0.28 mm, which is smaller than 0.43 mm. So it can be concluded that taking into account the spandrel wall lead to a more stiff model.

Figure 5-15 Maximum principle stress on the three dimensional bridge

Figure 5-16 Maximum principle stress on the arch




In Figure 5-15 and 5-16 show the maximum principle stress on the masonry bridge model and o the arch of it. In Figure 5-16 it can be seen clearly four places of stress concentration, where the masonry material exceeds its tensile strength. So four plastic hinges had been developed on the arch. They were located on extrados of arch on left foot; on intrados of arch under loading point; on extrados of arch on point symmetric to loading point; on intrados of arch on right foot.

Figure 5-17 Deformation on the arch under the loading point with load increment

Figure 5-18 Deformation on the arch at place symmetrical to loading point with load increment

Figure 5-17 and 5-18 shows the vertical displacement – load graph of the node on the arch under loading point and on the symmetric point to loading point. The point under the loading point is




descending while the point symmetric to it is rising. Before the load 3.5 kN, the structure was without elastic linear range, and after that load plastic nonlinear characteristic had been performed.

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Figure 5-19 Load-vertical displacement comparison between three dimensional model, plane stress model and experimental result at the point under loading point

Figure 5-19 shows the load-vertical displacement graph of three dimensional model, plane stress model and experimental result. The graph is obtained from the node on intrados of the arch under the loading point. From Figure 5-19 it can be seen the vertical displacement is smaller for three dimensional model than plane stress model in each load step. The plane stress model appears a softer behavior. It is because the three dimensional model took into account the spandrel wall. In masonry arch bridge, spandrel walls play a very important role in increasing the ultimate capacity of the structure. So it can be concluded that spandrel wall which was setup in three dimensional model increased the stiffness of the masonry arch bridge and as a result, make the ultimate capacity of the bridge higher. When compared with experimental result, both the plane stress model and three dimensional model played a smaller displacement in each load step. The finite element models were stiffer than the real experiment. This phenomenon may caused by the heterogeneous characteristic of material used in the experimental specimen, which cause early cracking unexpected as well as by sliding of the abutment which is not real fixed boundary condition during the experimental process.




6. SUMMARY AND CONCLUSIONS

6.1 Conclusions

This dissertation addresses the stimulation of experiment on masonry arch bridge, focusing on infill and masonry arch behavior with various material properties, structural characteristic and boundary conditions of the masonry arch bridge and performing numerical analysis using limit analysis software RING 1.5 as well as finite element method software DIANA. The preliminary step is very important to understand the experimental Champaign. It is necessary to be aware of the material utilized in the experiment, the real boundary conditions, the loading procedure and the results acquired from the experiment. Apart from that, a limit analysis is developed in this dissertation as a first stimulation, which has been proved to be a very easy execute and effective method to assess the ultimate capacity and collapse mechanism. Following that, plane stress finite element models with different number of soil layers, masonry tensile strength and lateral constrains are made to be better understand the influence for each of them to the structural behavior. Based on that, three dimensional finite element model is constructed in DIANA software. The three dimensional model consider the effect of spandrel wall and can be a better stimulation to the realistic experiment.

The main conclusions are commented below.

For the plane stress model:

Different masonry tensile strength leads to the result that how full the hinges on the arch develop. The less tensile strength the more fully development of the hinges on different area. And the low masonry tensile strength also causes more soft structure which has more deformation capacity.

From the comparison for model with one soil layer and model with three soil layers, the assumption that elastic modulus is not uniform along the depth but increase from top to bottom, is very close to the realistic experiment. The stiffness of three layers model are smaller than the one layer model. The ultimate load reached of the three soil layers model is similar to the result of using limit analysis method, which is 10% error. Regarding the difference on lateral boundary conditions masonry tensile strength, the error is rather small. And in this plane stress finite element model, the four hinges collapse mechanism is fully developed. The hinges formed on the intrados of arch under loading point, extrados of arch on symmetric point to the loading point and two feet of the arch.

By comparing the model with steel bracing system to the model without it, we can conclude that using steel bracing system lead us to obtain more realistic results. While modeling without steel




bracing could not develop the full four hinges collapse mechanism (only three hinges appear at the ultimate load), the steel bracing stimulation make it possible to check the four hinges obviously shown on the masonry arch. The stiffness is smaller in the model with steel bracing than in the model without it.

For the three dimensional model：

Three dimensional model finite element analyses are very complex and time-consuming. We need to be very careful when defining the geometry and material properties as well as meshing. After that, checking and calibrating model is necessary to avoid any unexpected mistake. Load steps deserve more attention; they should be small enough to achive convergence. The addition of the spandrel wall will increase the ultimate capacity of the masonry arch bridge model. The three dimensional model has more stiffness than the plane stress model. The experimental result indicated the test bridge has less stiffness than plane stress model and three dimensional model. It is because the heterogeneity of soil and masonry material used in the bridge and the unrealized completely fixed abutment during experimental process.

6.2 Suggestions for further study

Due to the time limitation, several possible trials that may improve the numerical stimulation could not be made. The future work to be done may include:

The finite element model with more soil layers could be made to stimulate in a more realistic situation for the masonry arch bridge. In this dissertation only three layers were designed. In the future work soil can be divided into many layers with different Young’s modulus.

The effect of mesh size had not been discussed in this dissertation. Because the finite element analysis, especially the three dimensional model analysis is very complex and time consuming. One major factor that influent the computaion time is the number of elements and nodes. It could be interesting to compare models with different mesh divisions to give better instruction in optimizing modeling process.

Due to the time limitation, comparison of models with different soil layers and boundary conditions were not able to be performed. So in the future, these factors which affect the analysis result could be studied, particularly the boundary conditions at the abutments.




Also, an accurate analysis of the ties' stresses will provide further insight on the model performed.




7. REFERENCES

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Mitsuo Tsutsui (2010), Line of thrust and theoretical load of masonry arch bridge, ARCH’10 – Proceedings of the 6th International Conference on Arch Bridges, Fuzhou, China.

F. Biondini (2002), Uncertainties in the nonlinear analysis of masonry bridge structures, Structural Safaty and Reliability, Corotis et al. (eds) © 2002 Swets & Zeitlinger, CA Lisse, ISBN 90 5809 197X.

Yu Zheng (2010), Nonlinear finite element analysis of masonry arch bridges reinforced with FRP, ARCH’10 – Proceedings of the 6th International Conference on Arch Bridges, Fuzhou, China.

Begimgil, M. (1995), Behavior of restrained 1.25 m model masonry arch span bridge. Arch Bridges. Proceedings of the first International Conference on Arch Bridges Held at Bolton. Edited by C. Melbourne. p. 321-326.

Arenas, J.J. (2001), Bridges in the landscape. OP Planning and Engineering Magazine, Vol 1, No. 54, p. 54-59.

Hughes, T.G. (1996), The testing, analysis of masonry arch and Assessment bridges. Structural Analysis of Historical Constructions. Possibilities of numerical and experimental techniques. Edited by: P. Roca, J.L. Gonzalez, A.R. Mari and E. Oñate, p.64-84. CIMNE.

Hughes, T.G., Blacker, M.J. (1997), A review of the UK masonry arch Assessment methods. Structures and Buildings. Proceedings of the Institution of Civil Engineers, 122, p.305-315.

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Molins, C., Roca, P. (1998), Load capacity of masonry arch bridges multi-. Arch Bridges. Anna Sinopoli, editor, p. 213-222. Balkema, Rotterdam.

Page, J. (1993), Masonry arch bridges. State of the Art Review. TRL. Department of Transport.

Page, J. (1995), Load tests on masonry arch to collapse bridges. Arch Bridges. Proceedings of the first International Conference on Arch Bridges Held at Bolton. Edited by C. Melbourne. p. 289-298.




Roca, P., Molins, C., Hughes, T.G., Sicily, C. (1998), Numerical simulations of experiments in arch bridges. Arch Bridges. Anna Sinopoli, editor, p. 195-204. Balkema, Rotterdam.

Soms, E. (2001), Anàlisi última de ponts arc d’obra de fàbrica. Thesis specialty ETSECCPB-UPC.

P. Roca, A. Marí, E. Oñate. (10996), The Testing, Analysis and Assessment of Masonry Arch Bridges' TH: Hughes. Structural Analysis of Historical Constructions CIMNE. 1996.

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